MHB How can we prove that a Pythagorean triple is primitive?

  • Thread starter Thread starter Poirot1
  • Start date Start date
  • Tags Tags
    Primitive
AI Thread Summary
To prove that a Pythagorean triple (x, y, z) = (a^2 - b^2, 2ab, a^2 + b^2) is primitive, it is essential to establish that gcd(x, y, z) = 1. This can be shown by ensuring that the integers a and b are coprime (gcd(a, b) = 1) and of opposite parity, meaning one is even and the other is odd. A contradiction argument can also be employed, assuming a common factor d greater than 1, leading to a prime factorization. The discussion emphasizes the importance of these conditions in confirming the primitiveness of the triple. Understanding these principles is crucial for mathematical proofs involving Pythagorean triples.
Poirot1
Messages
243
Reaction score
0
I know that (a^2-b^2,2ab,a^2+b^2) is pythagorean triple. How to show it is primitive? i.e
gcd(x,y,z)=1
 
Mathematics news on Phys.org
Re: primitive pythagorean triple

Hello, Poirot!

I know that (x,y,z) = (a2 - b2, 2ab, a2 + b2) is a Pythagorean triple.

How to show it is primitive? .i.e. gcd(x,y,z) = 1
I'm not sure how we "show" it, but here is a fact.

For a primitive Pythagorean triple, a and b must be of opposite parity.
. . (One must be even, the other must be odd.)
 
Re: primitive pythagorean triple

Poirot said:
I know that (a^2-b^2,2ab,a^2+b^2) is pythagorean triple. How to show it is primitive? i.e
gcd(x,y,z)=1
The conditions for the triple to be primitive are that gcd(a,b)=1 and a, b are of opposite parity. See Pythagorean triple - Wikipedia, the free encyclopedia.
 
Re: primitive pythagorean triple

I'm pretty sure a contradiction argument is expedient.

assume x,y,z have a common factor d not equal to 1.
Since everything can be factorized into primes can I assume d is prime?
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Similar threads

A Generalized Diophantine equation and the method of infinite descent
Replies
4
Views
1K
MHB Number of natural numbers that have primitive roots
Replies
5
Views
1K
I MathCan Numerical Observations Lead to a Simpler Proof of Fermat's Last Theorem?
Replies
20
Views
4K
I Equivalent definitons for primitive polynomials
Replies
24
Views
422
MHB Why Must One Leg of a Primitive Pythagorean Triple Be a Multiple of 3?
Replies
1
Views
2K
PE 75 primitive Pythagorean triples
Replies
37
Views
4K
I Fit Triples into Sixtuples in the best way?
Replies
7
Views
2K
Number theory, primitive pythagorean triples
Replies
6
Views
2K
B Pythagorean triples that sum to 60
Replies
2
Views
2K
I Questions Regarding Primitive Unit Cell
Replies
4
Views
4K

Hot Threads

Recent Insights

Back
Top